7 ALGEBRAIC PROPERTIES OF CONVOLUTION 277 (14.7.3) If the sequence (u^..., un) is convolvable, then so is the sequence (14.7.3.1) (//! * fi2 * • • • * uny = £in * #„_! * • • • * /V For (14.1.4 and 13.7.10) we have if and only if f-r j-r and these two integrals are equal. On the other hand, if the sequence (A, u) is convolvable, it does not necessarily follow that (/*, A) is convolvable (Problem 2). But if G is com- mutative this will be the case, and we shall have A * \JL = fj. * A. In particular, it follows from the preceding results that (14.7.4) On the set M£(G) of bounded measures on G, the law of composition (A, p) i-> A * IJL (together with the vector space structure) defines a C-algebra structure\ the unit element is the Dirac measure se at the neutral element e ofG. The set M£(G) of compactly supported measures on G is a subalgebra of Mc(G). The algebra M£(G) is commutative if and only ifG is commutative. The fact that G is commutative if M£(G) is commutative follows from the formula (14.6.1.2). If G is discrete, the algebra M£(G) consists of all linear combinations ]T as8&, where as = 0 for all but a finite number of points s e G (3.16.3), and seG the formula (14.6.1.2) shows that seG / \seG This is what is called in algebra the group algebra of the group G over the field C.f*4.(G), hence is (A * ^)-integrable, and we have